Polynomials over ℤ and ℚ: counting and freeness

Humans have been thinking about polynomial equations over the integers, or over the rational numbers, for many years. Despite this, their secrets are tightly locked up and it is hard to know what to expect, even in simple looking cases. In this talk I’ll discuss recent efforts to understand the frequency of integer solutions to cubic polynomials, before turning to the much more evolved picture over the rational numbers.